detB

In mathematics, detB is a common shorthand for the determinant of a square matrix B. The determinant is a scalar value that summarizes certain properties of the linear transformation represented by B, including how it scales volumes and whether it preserves orientation. A nonzero determinant indicates that the transformation is invertible.

If B is an n×n matrix, det(B) is the product of its eigenvalues (counting multiplicities). It equals zero precisely when B is singular (noninvertible). Fundamental algebraic properties include det(AB) = det(A) det(B); det(B^T) = det(B); det(kB) = k^n det(B) for any scalar k and n×n B; and det(B^{-1}) = 1/det(B) when B is invertible. For a 2×2 matrix B = [a b; c d], det(B) = ad − bc.

Computationally, determinants are typically calculated using LU decomposition, which factors B into lower and upper triangular matrices; the determinant is the product of the diagonal entries of the triangular factors, times a sign factor from row permutations. Direct cofactor expansion is impractical for large matrices. In numerical work, the log-determinant is often used to improve numerical stability.

Applications span solving linear systems, changing variables in calculus, and stability analysis in dynamical systems. The determinant also encodes volume: the absolute value of det(B) is the factor by which B scales n-dimensional volume, and det(B) equals the product of eigenvalues.

In other contexts, detB may refer to the determinant of a matrix representing a bilinear form, or may be used as a variable name; the precise meaning depends on the surrounding mathematical notation.